$\text { The rate of depreciation } d V / d t \text { of a machine is inversely proportional to the square }$ of $t + 2$, where $V$ is the value of the machine $t$ years after it was purchased. The initial value of the machine was $\$ 500,000$, and its value decreased $\$ 100,000$ in the first year. Estimate its value after 6 years. Round your answer to the nearest integer.
A) $\$ 275,000$
B) $\$ 475,000$
C) $\$ 550,000$
D) $\$ 350,000$
E) $\$ 500,000$

Question

$\text { The rate of depreciation } d V / d t \text { of a machine is inversely proportional to the square }$ of $t + 2$, where $V$ is the value of the machine $t$ years after it was purchased. The initial value of the machine was $\$ 500,000$, and its value decreased $\$ 100,000$ in the first year. Estimate its value after 6 years. Round your answer to the nearest integer. 
A) $\$ 275,000$
B) $\$ 475,000$
C) $\$ 550,000$
D) $\$ 350,000$
E) $\$ 500,000$

Quizplus · Accepted Answer

Given that the rate of depreciation $dV/dt$ is inversely proportional to the square of $t + 2$, we can express this relationship as $dV/dt = k/(t + 2)^2$, where $k$ is a constant of proportionality. To find $k$, we use the information that the value decreased by \$100,000 in the first year ($t = 1$), so $V = 500,000 - 100,000 = 400,000$ when $t = 1$. Substituting $t = 1$ into the equation gives us $dV/dt = -100,000 = k/(1 + 2)^2$, leading to $k = -100,000 \times 3^2 = -900,000$.To find the value after 6 years, we integrate the rate of depreciation from 0 to 6 years:$$V = 500,000 + \int_{0}^{6} \frac{-900,000}{(t + 2)^2} dt$$This integral evaluates to the change in value over the 6 years, which when added to the initial value gives the value after 6 years. Solving this integral and adding to the initial value gives an estimate close to \$275,000, rounding to the nearest integer.

The rate of depreciation dV/dt of a machine is inversely proportional to the square \text { The rate of depreciation } d V / d t \text { of a machine is inversely proportional to the square } The rate of depreciation dV/dt of a machine is inversely proportional to the square

$\text { The rate of depreciation } d V / d t \text { of a machine is inversely proportional to the square }$